the Pythagorean Comma

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The comma has its origin in the Pythagorean Tuning System, which is generated by 12 successive just (perfect) fifth intervals (ratio 2:3).
But the span of 12 just fifths exceeds the span of 7 octaves (each of a ratio 1:2) by a tiny bit - the Pythagorean Comma - leading to the necessity of tempered tuning systems.

As long as we stay in the world of drones and their harmonic overtones everything is relatively simple: From a mathematical perspective it is all a game with the first 20 integers or so and the proportions between them.
But as soon as we also include melody, the whole picture becomes more complex, and the most fundamental functions from the harmonic series appears not to meet precisely.
The most fundamental structure in music is the division of the octave (ratio 1:2) in twelve halftones.
In the modern equal temperament all halftone steps are mathematically equal but in this arrangement the just intervals have been sacrificed - specially the thirds and sixths.
The reason for this is that the structure originally derrives from the primary elements of the harmonic series, the octave, 1:2, and the perfect fifth, 2:3.
The span of 7 octaves - 2 to the power of 7 - is almost the same as the span of 12 perfect fifths - 3/2 to the power of 12 - but if you calculate the values you will come to 128 and 129.746 respectively.
So something needs to give way when the 12 tones are arranged within the frames of an octave in a tonal system.
Like the sisters of Cinderella musicians around the globe have been cutting the heels and toes of the musical intervals to make them fit the glass (or gold) slipper - the illusion of a perfect tonal system.
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